# What is the history behind the concept of “schemes” in algebraic geometry?

Let me start by saying that I've just started studying algebraic geometry. I have a habit of trying to find out something about the motivation behind new concepts that I'm studying, in this case that means that I'm trying to find out something about how the concept of schemes arose in algebraic geometry, and how to view schemes from a historical perspective.

• The question is answered here hsm.stackexchange.com/questions/1813/… – Conifold Aug 15 '15 at 22:44
• Two recommended references: "The Rising Sea: Grothendieck on simplicity and generality" by Colin McLarty and an article (or the book if you're interested in the mathematical details) by Dieudonné, from 1972 (respectively 1985), called The Historical Development of Algebraic Geometry, The Amer. Math. Monthly, Vol. 79, No. 8, pp. 827-866. (The book is called History of algebraic geometry.) – Ben Aug 16 '15 at 7:33

“A story says that in a Paris café around 1955 Grothendieck asked his friends “what is a scheme?”

The very first time the word “schéma” was uttered, in Paris, at an official seminar talk, was during the Cartan seminar of 1955/56 on algebraic geometry.

The lecturer was Claude Chevalley, and the date was december 12th 1955.

http://www.neverendingbooks.org/grothendiecks-cafe

The rising sea, Mclarty

The history is simple:-) Schemes were invented by Grothendieck. The purpose was unification and simplification of the foundations of algebraic geometry.

The general idea is to shift from considering points of a manifold to certain ideals of a ring of functions on this manifold. This general idea evolved gradually. An early predecessor was Gelfand's theory of commutative Banach algebras, where he starts with an algebra and recovers a space so that this algebra is the algebra of functions on this space. The points of the space are defined as maximal ideals. Grothendieck did similar thing in algebraic geometry. He proposed to consider prime ideals of a ring as points of certain space.

For the reference on Grothendieck's original text see "Elements de geometrie algebrique" on Wikipedia. I believe it was never completely published but various versions were widely circulated. So there is no well-defined publication year. Grothendieck worked and reworked it for almost 10 years ending around 1970.

• I think that it would be a good idea to expand a bit on the historical aspect of this answer. The mathematical content is already nicely covered. Could you provide some references to papers, or review articles discussing this? – Danu Aug 15 '15 at 14:08
• I think this is mentioned in one of the Grothendieck biographies, but I have to check. This concerns the influence of Gelfand. Grothendieck's own publications are well known: EGA and SGA. – Alexandre Eremenko Aug 16 '15 at 2:57
• Okay, well please update your answer once you find out more about the details :) – Danu Aug 16 '15 at 7:32
• No, schemes were not invented by Grothendieck. Chevalley gave a presentation at Cartan's seminar titled Les schémas in 1955, while Grothendieck was still in Kansas doing algebraic topology, and learning classical algebraic geometry with the help of Serre's letters. Chevalley was developing ideas of Zariski. EGA only appeared starting in 1960. – Conifold Sep 23 '16 at 21:41

"The term itself was coined by Chevalley, although accepted in a more restrictive sense than the term as used by Grothendieck. In Foundations of Algebraic Geometry, André Weil had introduced into algebraic geometry the methods used by his mentor, Élie Cartan, in differential geometry (following Carl Friedrich Gauss and Jean Darboux). But Weil’s method was by no means intrinsic, and Chevalley wondered what was invariant, in Weil’s sense of variety. The answer, inspired by Zariski’s work, was simple and elegant: the scheme of an algebraic variety is the collection of local rings of the sub-varieties found inside the rational function field. There is no need for an explicit topology, a point of distinction between Chevalley and Serre, who at roughly the same time introduced his algebraic varieties using Zariski topologies and sheaves. Each of the two approaches had advantages, but also limitations: Serre had an algebraically closed base field; Chevalley had to work only with irreducible varieties. In both cases, the two fundamental problems of products of varieties and base change could only be approached indirectly. All the same, Chevalley’s point of view was better suited to future extensions to arithmetic, as Nagata soon observed.

Évariste Galois was certainly the first to notice the polarity between equations and their solutions. One must distinguish between the domain in which coefficients of the algebraic equation are chosen and the domain in which solutions are sought. Grothendieck created a synthesis out of these ideas, based in essence on the conceptual presentation of Zariski-Chevalley-Nagata. Schemes are thus a way of encoding systems of equations as well as the transformations to which one may subject them.

Grothendieck presented the Galois problem in the following manner: a scheme is an absolute object, X, say; the choice of a field of constants (or a field of definition) corresponds to the choice of another scheme S and a morphism πX from X to S. In the theory of schemes, a commutative ring is identified with a scheme, its spectrum. A homomorphism from ring A to ring B likewise maps, inversely, the spectrum of B into the spectrum of A. Moreover, the spectrum of a field has a single underlying point (even though many different points exist, in this sense); consequently, giving the field of definition as being included in the universal domain corresponds to giving a scheme morphism πT from T to S. A solution of the system of equations X, with the domain of constants S, with values in the universal domain T, corresponds to a morphism φ from T to X such that πT is the composition of φ and of πT."

A country known only by name, Cartier